Conceptual Proof of Quadratic Reciprocity

Question

What is a conceptual proof of quadratic reciprocity? I saw the wiki proof using algebraic machinery. I don't know what field extensions, Galois groups etc are. But I want to understand how properties under different moduli primes are getting related. I always wanted to understand how can one use all the modulo p information to understand globally. I would like to know if someone can explain quadratic reciprocity in a way that I will understand.

Answer 1

I don't think that there is any answer that is both elementary (comprehensible without knowing what fields, groups, etc are) and conceptually clear. There are combinatorial proofs such as by Zolotarev (cf. Zolotarev's lemma) or the one (by Eisenstein? or Gauss?) that counts lattice points under the line from (0,0) to (p,q), but they hardly dispel the mystery.

To "understand how can one use all the modulo p information to understand globally", there seems to be no way around the interpretation of QR using Hilbert symbols, or else the far more demanding approach using class field theory. In the Hilbert symbol formulation, there is a single object, the quadratic polynomial $Q(x,y,z) = x^2 - py^2 - qz^2$ that has nonzero solutions modulo any prime $k$ other than $p$ or $q$; has mod $p$ and mod $q$ solutions according to the signs of the Legendre symbols; and obeys a product formula stating that the product of all the $\pm$ signs at all primes is equal to $1$. This is a global constraint on the local solvability at different primes. The proof of the product formula is of course more complicated than the elementary proofs of quadratic reciprocity (you can find it in many places such as Serre's "course in arithmetic") but it has the advantage of being a general principle that applies to other situations.

As you can imagine, the one-sentence description of the product formula leaves out some details. Solutions mod $p$ are not enough, one needs solutions mod $p^n$ for all $n$, or equivalently, p-adic solutions, but for odd primes and for $Q(x,y,z)$ of degree 2 (i.e., less than $p$) this comes to the same thing. Real solutions are included as an "infinite" (Archimedean) prime in the product formula. The 2-adic and real "$\infty$-adic" solutions have to be treated separately as reflected in the existence of supplementary formulae for the Legendre symbols determining squareness of $2$ and $-1$ modulo odd primes. The basic point is the same, that there is a $\pm 1$ sign for each prime and a constraint that the product of all the signs is $+1$, something that is true for more general quadratic forms and for this particular $Q$ reduces to the statement of quadratic reciprocity.

The same story can be told in increasingly general versions for number fields, global fields and Riemann surfaces, rendered as a localization statement in K-theory, connected to the class field theory of quadratic fields, (maybe expressed in the language of motives), etc. For myself, I don't see how even with this additional 200 years' hindsight and extensive additional theory, the quadratic reciprocity formula is any less astounding than it was in Euler and Lagrange's time.

Answer 2

See Davenport "The Higher Arithmetic" for an elementary approach. Also see "An Introduction to the Theory of Numbers" for several approaches to proving quadratic reciprocity.